3.872 \(\int \frac{e^{\tanh ^{-1}(a+b x)}}{1-a^2-2 a b x-b^2 x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{\sqrt{a+b x+1}}{b \sqrt{-a-b x+1}} \]

[Out]

Sqrt[1 + a + b*x]/(b*Sqrt[1 - a - b*x])

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Rubi [A]  time = 0.0371005, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {6164, 37} \[ \frac{\sqrt{a+b x+1}}{b \sqrt{-a-b x+1}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a + b*x]/(1 - a^2 - 2*a*b*x - b^2*x^2),x]

[Out]

Sqrt[1 + a + b*x]/(b*Sqrt[1 - a - b*x])

Rule 6164

Int[E^(ArcTanh[(a_) + (b_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(c/
(1 - a^2))^p, Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]
&& EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a+b x)}}{1-a^2-2 a b x-b^2 x^2} \, dx &=\int \frac{1}{(1-a-b x)^{3/2} \sqrt{1+a+b x}} \, dx\\ &=\frac{\sqrt{1+a+b x}}{b \sqrt{1-a-b x}}\\ \end{align*}

Mathematica [C]  time = 0.095932, size = 12, normalized size = 0.44 \[ \frac{e^{\tanh ^{-1}(a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a + b*x]/(1 - a^2 - 2*a*b*x - b^2*x^2),x]

[Out]

E^ArcTanh[a + b*x]/b

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Maple [A]  time = 0.032, size = 42, normalized size = 1.6 \begin{align*} -{\frac{ \left ( bx+a+1 \right ) ^{2} \left ( bx+a-1 \right ) }{b} \left ( -{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/(-b^2*x^2-2*a*b*x-a^2+1),x)

[Out]

-(b*x+a+1)^2*(b*x+a-1)/b/(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)

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Maxima [B]  time = 1.51165, size = 286, normalized size = 10.59 \begin{align*} \frac{b^{2}{\left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{3} x + a b^{2} + \sqrt{b^{2}} b} - \frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{3} x + a b^{2} - \sqrt{b^{2}} b} - \frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{\sqrt{b^{2}} b^{2} x + a \sqrt{b^{2}} b + b^{2}} - \frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{\sqrt{b^{2}} b^{2} x + a \sqrt{b^{2}} b - b^{2}}\right )}}{2 \, \sqrt{a^{2} b^{2} -{\left (a^{2} - 1\right )} b^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/(-b^2*x^2-2*a*b*x-a^2+1),x, algorithm="maxima")

[Out]

1/2*b^2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(b^3*x + a*b^2 + sqrt(b^2)*b) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)
/(b^3*x + a*b^2 - sqrt(b^2)*b) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(sqrt(b^2)*b^2*x + a*sqrt(b^2)*b + b^2) -
sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(sqrt(b^2)*b^2*x + a*sqrt(b^2)*b - b^2))/sqrt(a^2*b^2 - (a^2 - 1)*b^2)

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Fricas [A]  time = 1.66461, size = 77, normalized size = 2.85 \begin{align*} -\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{2} x +{\left (a - 1\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/(-b^2*x^2-2*a*b*x-a^2+1),x, algorithm="fricas")

[Out]

-sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(b^2*x + (a - 1)*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/(-b**2*x**2-2*a*b*x-a**2+1),x)

[Out]

-Integral(1/(a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - sqrt(-a**2
- 2*a*b*x - b**2*x**2 + 1)), x)

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Giac [A]  time = 1.20698, size = 54, normalized size = 2. \begin{align*} \frac{2}{{\left (\frac{\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b}{b^{2} x + a b} - 1\right )}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/(-b^2*x^2-2*a*b*x-a^2+1),x, algorithm="giac")

[Out]

2/(((sqrt(-(b*x + a)^2 + 1)*abs(b) + b)/(b^2*x + a*b) - 1)*abs(b))